01/01/02deutsche bank: volatility copyright (c) 1997-2002, marshall, tucker & associates, llc....

01/01/02 Deutsche Bank: Volatility Copyright (C) 1997-2002, Marshall, Tucker & Associates, LLC. All rights reserved.

1

Deutsche Bank

Understanding Volatility

Alan L. Tucker, Ph.D.631-331-8024 (tel)631-331-8044 (fax)

[email protected]

Copyright © 1997-2002Marshall, Tucker & Associates, LLC

All rights reserved


2

ALAN L. TUCKER, Ph.D.

Alan L. Tucker is Associate Professor of Finance at the Lubin School of Business, Pace University, New York, NY and an Adjunct Professor at the Stern School of Business of New York University, where he teaches graduate courses in derivative instruments. Dr. Tucker is also a principal of Marshall, Tucker & Associates, LLC, a financial engineering and derivatives consulting firm with offices in New York, Chicago, Boston, San Francisco and Philadelphia. Dr. Tucker was the founding editor of the Journal of Financial Engineering, published by the International Association of Financial Engineers (IAFE). He presently serves on the editorial board of Journal of Derivatives and the Global Finance Journal and is a former associate editor of the Journal of Economics and Business. He is a former director of the Southern Finance Association and a former program co-director of the 1996 and 1997 Conferences on Computational Intelligence in Financial Engineering, co-sponsored by the IAFE and the Neural Networks Council of the IEEE.

Dr. Tucker is the author of three books on financial products and markets: Financial Futures, Options & Swaps, International Financial Markets, and Contemporary Portfolio Theory and Risk Management (all published by West Publishing, a unit of International Thompson). He has also published more than fifty articles in academic journals and practitioner-oriented periodicals including the Journal of Finance, the Journal of Financial and Quantitative Analysis, the Review of Economics and Statistics, the Journal of Banking and Finance, and many others.

Dr. Tucker has contributed to the development of the theory of derivative products including futures, options and swaps, and to the theory of international capital markets and trade. He has also contributed to the theory of technology adoption over the life-cycle. The Social Sciences Citation Index shows that his research has been cited in refereed journals on over one hundred occasions.

As a consultant, Dr. Tucker has worked for The United States Treasury Department, the United States Justice Department, Morgan Stanley Dean Witter, Union Bank of Switzerland, LG Securities (Korea), and Chase Manhattan Bank. Dr. Tucker holds the B.A. in economics from LaSalle University (1982), and the MBA (1984) and Ph.D. (1986) in finance from Florida State University. He was born in Philadelphia in 1960, is married (Wendy) and has three children (Emily, 1993, Michael and Matthew, both 1995).


3


Purpose. The purpose of this material is to impart a richer understanding of the measurement of and determinants affecting the volatility of equity returns. Topics covered include:

– Computing Implied Volatility Matrices

– Implied Volatility Term Structures

– Implied Volatility Skews and Smiles

– Forward Implied Volatility

– Computing Historic Volatility

– Historic Volatility Term Structures


4

It should be kept in mind that both historic and implied volatility are generally poor predictors of actual future volatility. This should be expected. It is a tautology that volatility is difficult to predict out-of-sample. After all, if it was readily predictable then it would not be volatility. “Predictable volatility” is an oxymoron.



5

Building an Implied Volatility Matrix.

CSCO Calls, Closing Prices, 28 February 2001

Strike Mar (16) Apr(41) July(132) October(223)

17.50 2 1/8 3 1/8 6 6 1/4

20.00 1 1/16 1 5/8 3 7/16 5

22.50 3/16 9/16 2 1/4 2 3/4

25.00 1/16 5/16 1 9/16 2 1/8

27.50 NA 1/8 15/16 1 1/2

30.00 NA NA 9/16 1

Numbers in parentheses represent days until option expiration. CSCO closing price on 28 February 2001 was $19.25. The interest rate term structure was essentially flat at 4.5% with continuous compounding. A calendar year is used.



6

Implied Volatility Matrix

Strike March April July October

17.50 64.46% 85.14% 114.76% 91.49%

20.00 84.98 74.03 78.84 86.12 (term structure)

22.50 65.63 60.83 71.28 61.88

25.00 73.25 67.23 69.80 62.43

27.50 NA 66.54 65.10 60.14

30.00 NA NA 62.44 57.41

(skew)

Example: 74.03% obtained by entering S = 19.25, X = 20.00, T - t = .1123 years (41 days), C = 1.625, and solving for implied volatility.



7

• Term structure effects are principally occasioned by:

– Mean Reversion

– Scheduled Informational Events

– Market Segmentations



8

Mean Reversion

Implied volatility tends to be mean-reverting (see J. Stein, J. of Finance, 1989), meaning that while it moves randomly in the short run it tends to be “pulled” to some long-run average level.



9

Scheduled Informational Events

Example: Recently a listed, one-day at-the-money call on IBM was trading at 3-3.125, resulting in an implied vol of over 400% per annum. IBM was scheduled to release earnings after the bell. Scheduled informational events and Merton’s 1973 finding that implied vol is a time-weighted average of spot and forward implied vols may be what underlies Stein’s finding of mean-reversion. (Discuss.)



10

Market Segmentations

These are “pockets” of excess demand or supply, which affect option prices and therefore implied vols. For example, there may be excess demand for long-term, deep out-of-the-money index put options by institutional money managers seeking to purchase disaster insurance, resulting in somewhat higher implied vols for these options vis-à-vis their shorter-term counterparts. Index puts expiring near the end of calendar quarters also often exhibit higher implied vols.



11

• Skews and Smiles are most often occasioned by:

– Violations of the Assumption that Prices are Log-normally

Distributed

– Leverage

– Market Segmentation

– Option Maturity



12

Violations of the Assumption that Prices are Log-normally Distributed.

Stock and stock index options most often exhibit an implied volatility skew whereby implied volatility decreases as the strike price increases. (This is true for our CSCO matrix.) Thus deeper out-of-the-money puts and deeper in-the-money calls have significantly higher implied vols. A price distribution that has more kurtosis and a fatter (thinner) left (right) tail than a log-normal distribution with the same mean and standard deviation, is consistent with this type of skew. Empirical evidence suggests that the implied probability distribution for a stock price has fatter left tails than the probability distribution calculated from empirical data on stock market returns. Rubinstein (J. of Finance, 1994) refers to this as “crashophobia”. On the other hand, currency options tend to exhibit implied vol smiles - consistent with a price distribution that has fatter left- and right-hand tails than a log-normal with the same mean and standard deviation. Perhaps this is due to the possibility of Fed intervention policies and exchange rate realignments by foreign governments.



13

Leverage.

One possible explanation for the skew in equity options concerns leverage. As a company’s equity declines in value, the company’s leverage increases so the equity becomes more risky and its volatility increases, and vice-versa. Here the volatility of equity would be a decreasing function of price. This may be regarded as a type of “level effect” (which is often seen in interest rates whereby interest rate volatility is directly related to interest rate levels in the economy), and is the motivation behind the CEV model of Cox and Ross (J. of Financial Economics, 1976).



14

Market Segmentation.

Again, pockets of excess demand or supply can occasion premium differences and thus implied volatility smiles and skews for options with the same maturity but different moneyness.



15

Option Maturity

The degree of the smile or skew tends to dampen with option maturity. (See our CSCO data, for example. Also see Natenberg, Option Pricing and Volatility: Advanced Trading Strategies and Techniques, 2nd ed., Chicago: Probus, 1994.)



16

• On the Role Played by the Pricing Model.

An oft asked question is, how important is the pricing model selected (Black-Scholes versus CEV versus Jump-Diffusion) if traders are prepared to use a different volatility for every deal (different maturity, different strike, put or call)? It can be argued that an option pricing model is no more than a tool used by traders for understanding the volatility environment and for pricing illiquid securities consistently with the market prices of actively traded securities. If traders stopped using Black-Scholes for example, and switched to another plausible model, then the volatility matrix would change and the shape of the term structure, smile and skew would change. But arguably the prices quoted in the market would not change appreciably.



17

Forward implied volatility is computed from spot implied volatility much like a forward interest rate is computed from spot rates. Keep in mind however that option volatility refers to the annualized standard deviation of the continuously compounded rate of return of the underlying asset, whereas most forward rate computations involving interest rates involve a different compounding frequency (such as a semi-annual periodicity in the US Treasury bond market).

Question: What is the forward implied volatility of the nearest at-the-money CSCO call between the third Friday of July and the third Friday of October?



18

Answer: The two relevant spot implied volatility measures are those of the CSCO July 20 and CSCO October 20 calls. These are 78.84% and 86.12% respectively. Recall there are 132 days until the July expiry and 223 days until the October expiry, and thus we are looking for an estimate of the 91-day forward vol that begins 132 days hence. Call this FV. The answer is obtained as follows:

exp(.8612)(223/365) = exp(.7884)(132/365) x exp(FV)(91/365)

(.8612)(223/365) = (.7884)(132/365) + (FV)(91/365)

FV = .9668 = 96.68%.

Note that, just like with interest rate term structures, if the relevant segment of the spot implied volatility term structure is upward sloping, then the forward implied volatility term structure that it begets is even more steeply upwardly sloped. Also note that it is possible to compute other forward implied vols and therefore entire forward implied volatility term structures. These may be useful, for example, for pricing forward-starting options such as executive stock options that have a vesting period.



19

Measuring Historic Volatility. There are many ways to measure volatility using historic data. We will focus on methods originally introduced by Engle (Econometrica, 1982) and used today in applications such as JP Morgan’s RiskMetrics.

Define V(n) as the variance of a stock, stock index, or other market variable on day n, as estimated at the end of day n - 1. Let SD(n) be its square root; SD(n) is commonly called “vol”.

Let the market variable at the end of day i be S(i). Let the variable U(i) denote the continuously compounded return during day i (between the end of day i - 1 and the end of day i):

U(i) = ln[S(i)/S(i - 1)].



20

A model that estimates V(n) while giving more weight to more recent data is:

(1) V(n) = E{a(i)[U(n - i)^2]},

where E is a summation operator where i = 1 to m, m is the total number of observations (sample size) of the daily U(i), and the variable a(i) represents the amount of weight given to the observation i days ago. The a’s are positive and a(i) < a(j) when i > j (because we want to assign less weight to older observations). The weights must sum to one, that is, E[a(i)] = 1.



21

An extension of equation (1) is obtained by assuming that there is a long-run average volatility and that this should be given some weight:

(2) V(n) = b(LV) + E{a(i)[U(n - i)^2]},

where LV is long-run volatility and b is the weight assigned to LV. (Now the sum of b and the a’s must be one.) Equation (2) is known as an ARCH(m) model, where the acronym ARCH stands for “AutoRegressive Conditional Heteroscedasticity”. In practice, b(LV) is replaced by a single variable, say w, when parameters are estimated.



22

The exponentially weighted moving average (EWMA) model is a particular case of equation (1) where the weights, a(i) decrease exponentially as we move back through time. Specifically, a(i + 1) = k[a(i)] where k is a constant between zero and one. This weighting scheme occasions the following simple formula for updating volatility estimates:

(3) V(n) = k[V(n - 1)] + (1 - k)[U(n - 1)^2].

Here the estimate of the variance for day n, V(n), which is made at the end of day n - 1, is calculated from V(n - 1) (the estimate that was made one day ago of the variance for day n - 1) and U(n - 1) (the most recent observation on changes in the market variable).



23

For example, suppose that k is 0.94, which is precisely the value that JP Morgan uses to update daily volatility estimates in its RiskMetric database. This value, being so close to 1, produces estimates of daily volatility that respond relatively slowly to new information provided by the U(i)^2. Also suppose that the volatility estimate for day n - 1 is 1% per day, and that the proportional change in the market variable during day n - 1 is 2%. So V(n - 1) = (0.01)^2 = .0001 and U(n - 1)^2 = (0.02)^2 = .0004. Equation (3) gives V(n) = 0.94 x .0001 + 0.06 x .0004 = .000118. The estimate of volatility is therefore (0.000118)^(0.50) = 0.01086278, or about 1.086% per day, or about 17.24% per annum using a trading day year (252 days).



24

A GARCH(1,1) model is a Generalized version of the EWMA model - itself a particular ARCH model - just described (see Bollerslev, J. of Econometrics, 1986). Here V(n) is calculated from a long-run average variance, LV, as well as from V(n - 1) and [U(n - 1)^2]. The model is:

(4) V(n) = b(LV) + a[U(n - 1)^2] + c[V(n - 1)].

Here c is a weight assigned to V(n - 1) and now a, b and c must sum to one. The EWMA model is a nested version of the GARCH model of equation (4) where b = 0, a = (1 - k) and c = k. [Under GARCH(1,1), V(n) is based on the most recent observation of U^2 and V. A more general GARCH(p,q) model calculates V(n) from the most recent p observations on U^2 and the most recent q observations of V. For asymmetric GARCH models and other variants, see Nelson (Econometrica, 1990) and Engle and Ng (J. of Finance, 1993).]



25

To accommodate parameter estimation, the term b(VL) is usually replaced by a single parameter w. Once w, a and c have been estimated, b is given by 1 - a - c. The long-term variance LV is then calculated as w/b. For example, suppose that a GARCH(1,1) model is estimated from daily data (using maximum likelihood estimation or variance targeting techniques, c.f. Engle and Mezrich, RISK, 1996) as:

V(n) = .000002 + 0.13 x U(n - 1)^2 + 0.86 x V(n - 1).

This corresponds to w = .000002, a = 0.13, c = 0.86, b = 0.01 and LV = 0.0002. In other words, the long-run average variance per day is 0.0002, corresponding to a vol of 1.4% per day. Now suppose that the estimate of the vol on day n- 1 is 1.6% per day so that V(n -1) = 0.000256 and that the proportional change in the market variable on day n - 1 is 1% so that [U(n - 1)]^2 is 0.0001. Then:

V(n) = 0.000002 + 0.13 x 0.0001 + 0.86 x 0.000256 = 0.00023516.

The new estimate of the volatility is therefore (0.00023516)^(0.50) = 0.0153 or

1.53% per day.



26

Recall that the GARCH(1,1) model is similar to the EWMA model except that, in addition to assigning weights that decline exponentially to past U^2, it also assigns some weight to the long-run average volatility. Because of this added feature, GARCH models can accommodate mean reversion in the volatility. Indeed, the parameter c in the model is a type of “decay rate” similar to the parameter k in the EWMA model. And for GARCH(1,1) per se, the variance V(n) exhibits mean reversion with a reversion level of LV and a reversion rate of 1 - (a + c). (In the EWMA model, (a + c) = 1 so the reversion rate is zero.)

[Also recall that Stein found that implied volatility tends to be mean reverting - itself following a type of GARCH model. This finding suggests some trading strategies involving implied volatility that are described in more detail in the moduel titled “Common Option Strategies”. For example, being “long vol” when implied volatility is below its long run average level can prove profitable if the reversion rate is relatively fast. An option book that is more or less delta and gamma neutral but has a positive vega will profit here. If the implied vol is significantly above its long run average level, then a dealer may want to “short vol” by being a net writer of options and then attempting to capture value by delta hedging where the deltas are computed using the lower long-run volatility estimate. In other words, the dealer sells options at a high vol but bases her hedging strategy on a lower vol. This will prove profitable if indeed the actual vol proves to comport more with the long-run/lower vol.]



27

• Building an Historic Volatility Term Structure. Because GARCH models accommodate mean reversion, they can be used to build forecasts of entire volatility term structures. These are call “historic volatility term structures” because they are built using historic data samples and are therefore not to be confused with implied volatility term structures. Still, one might already envision trading strategies based on comparisons of the two term structures.

It can be shown that the GARCH(1,1) model described in equation (4) occasions the following estimate of future variance:

(5) V(n + f) = LV + {[(a + c)^f] x (V(n) - LV)}

where V(n + f) is an estimate of the variance to occur on day n + f in the future. Notice that when a + c < 1, the final term in equation (5) becomes progressively smaller as f increases. (If a + c > 1then the variance would not be reverting but would be “fleeing”.)



28

To build a historic volatility term structure suitable for options, consider an option that lasting between day n and day n + N. One can use equation (5) to compute the expected variance during the life of the option as:

(6) (1/N) x S[V(n + f)],

where S is a summation operator for f = 0,…,N - 1.



29

Suppose that for a particular stock, a + c = 0.9602 and LV = 0.00004422 (a long-run daily vol of 0.66498%, or a trading-day yearly vol of 10.556%). Also suppose that the current variance per day, V(n), is 0.00006. This corresponds to a daily vol of 0.77460%, which is greater than LV, and an annual vol of 12.30% based on a 252 trading-day year. The first table below shows the historic volatility term structure (% per annum for a calendar day year) based on these data and equation (6), while the second table shows the impact on the term structure of a 1% change in the instantaneous volatility. Of course, once one has a volatility term structure like that in the first table below, one can always compute a forward vol and indeed an entire forward vol term structure a la getting a forward implied vol from an implied volatility term structure.

Notice in first table how the vol predicted in 500 days is closing in on the LV value of 10.556%. Also notice that the term structure is downward sloping, so the forward term structure would be even more steeply downwardly sloped.



30

Option Life (Days) 10 30 50 100 500

Option Volatility 12.03 11.61 11.35 11.01 10.65

(% per annum, 252-day trading year)

Option Life (Days) 10 30 50 100 500

Option Volatility Now 12.03 11.61 11.35 11.01 10.65

After 1% Change 12.89 12.25 11.83 11.29 10.71

Increase in Volatility 0.86 0.64 0.48 0.28 0.06

Note: 12.30% current daily vol moves to 13.30%. So the daily vol becomes 0.84% and the daily variance becomes 0.00007016. Applying equation (6) to this new situation produces the second row in the second table.


01/01/02deutsche bank: volatility copyright (c) 1997-2002, marshall, tucker & associates, llc....

Documents

tucker associates

journal of finance

financial futures

financial products

journal of economics

global finance journal

volatility copyright

journal of banking